| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 7 |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring standard techniques: using sum of probabilities equals 1 and expectation formula to find two unknowns (part a), then calculating variance (part b) and linear transformation of expectation (part c). All steps are routine applications of formulas with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Score | 0 | 1 | 2 | 3 |
| Probability | \(a\) | \(b\) | \(b\) | \(b\) |
| Answer | Marks |
|---|---|
| M1, A1 | www [NB: a not needed] |
| Answer | Marks |
|---|---|
| M1, M1, A1 | (no additional guidance) |
| Answer | Marks |
|---|---|
| M1, A1 | Or £0.70 but not £0.7; Accept -70p oe |
## (a)
$0a + b + 2b + 3b = 0.9$
$b = 0.15$ AG
| M1, A1 | www [NB: a not needed] |
## (b)
$0a + 1b + 4b + 9b$
$-0.9^2$
$= 1.29$
| M1, M1, A1 | (no additional guidance) |
## (c)
$E(2X - 2.5) = -0.7$
Expected loss is 70p
| M1, A1 | Or £0.70 but not £0.7; Accept -70p oe |
---2 In a fairground game a competitor scores $0,1,2$ or 3 with probabilities given in the following table, where $a$ and $b$ are constants.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Score & 0 & 1 & 2 & 3 \\
\hline
Probability & $a$ & $b$ & $b$ & $b$ \\
\hline
\end{tabular}
\end{center}
The competitor's expected score is 0.9 .
\begin{enumerate}[label=(\alph*)]
\item Show that $b = 0.15$.
\item Find the variance of the score.
\item The competitor has to pay $\pounds 2.50$ to take part, and wins a prize of $\pounds 2 X$, where $X$ is the score achieved. Find the expectation of the competitor's loss.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q2 [7]}}